If $x$ is transcendental and $y$ is algebraic, what about $x+y$ and $xy$?

Question

I have a field extension $E/K$ and $x,y\in E$ where $x$ is transcendental and $y$ is algebraic. I want to know whether $x+y$ and $xy$ are algebraic or transcendental. I do not know how to begin.

Answer 1

If $y\neq0$ is algebraic then so are $-y$ and $y^{-1}$. So if either $x+y$ or $xy$ is algebraic then $$x=x+y+(-y)\qquad\text{ or }\qquad x=xy(y^{-1}),$$ is algebraic, a contradiction.

Answer 2

Hint: the set of all algebraic elements (over some given field) is a field.